Conformal Group Actions on Generalized Kuramoto Oscillators
aa r X i v : . [ m a t h . D S ] D ec Conformal Group Actions on Generalized Kuramoto Oscillators
Max LiptonCornell UniversityDecember 19, 2018
Abstract
This paper unifies the recent results on generalized Kuramoto Model reductions. Lohe took a coupledsystem of N bodies on S d governed by the Kuramoto equations ˙ x i = Ω x i + X − h x i , X i x i and usedthe method of Watanabe and Strogatz to reduce this system to d + d ( d − equations. Using a model ofrigid rotations on a sphere as a guide, we show that the reduction is described by a smooth path in theLie group of conformal transformations on the sphere, which is diffeomorphic to SO ( d ) × D d . Seeingthe reduction this way allows us to apply geometric and topological reasoning in order to understandqualitative behavior of the Kuramoto Model.keywords: Kuramoto Model, dynamical systems, Lie groups, conformal geometry Suppose we mark N points on a sphere S d and rotate it over time via a continuous path R ( t ) in SO ( d + 1),tracking the location of the individual points. Noether’s Theorem tells us there is a system of conservedquantities, and it is readily apparent the angles between the points (in R d +1 from the standard sphereembedding) are conserved over time. Rather than expressing the position of point x i based on the givenrotations of the sphere, suppose we start with the N coupled differential equations for the points and wededuce the pairwise angles between points, or some equivalent quantity like the pairwise inner products, areconserved. Then from this information, we can derive a single differential equation ˙ R = f ( R ) in SO ( d + 1),dramatically reducing the number of variables. The Kuramoto Model follows this footprint, except theconserved quantities are the cross ratios preserved by conformal transformations.Rennie Mirollo connected the classical model on S to the Mobius group of fractional linear transformationson R , a topic which is well-understood [3]. Max Lohe derived reduced equations in higher dimensionalKuramoto models [5], whilst Chandra, Girvan, and Ott derive the higher dimensional continuum limit onthe number of bodies [2]. The conserved quantities Lohe derived are those that are preserved by conformaltransformations on the sphere. We have tied up these disparate results into a unified statement aboutconformal dynamical systems, and hopefully we can use this interpretation to get a concrete qualitativestatement on higher dimensional Kuramoto Models. 1 Classical Kuramoto Results
We denote the d -dimensional sphere by S d , and we denote the d -dimensional open disc by D d .The Kuramoto Model is a well-studied collection of dynamical systems which model coupled oscillators.Depending on the choice of coupling and its parameters, phenomena such as synchronization, segregation,and scattering can occur. Kuramoto Models have a diversity of applications including population sleepcycles, Josephson junctions, systems of mechanically linked motors, and more.The classical one dimensional case concerns a system of N coupled oscillators on S ⊆ C , written as˙ θ j = A + B cos θ j + C sin θ j , j = 1 . . . N , where A, B, and C are smooth functions of time and the angularvariables θ , . . . , θ N . For many of the preexisting results on conserved quantities to hold, we require thatthe coefficient functions are the same for all j and they involve all the angular variables. Sometimes thesystem will be expressed in a form that obfuscates A, B, and C , but by some clever use of trigonometricidentities, we can rewrite the governing equations in the desired form. For example, the first momentsystem ˙ θ j = ω + N N P i =1 e i ( θ i − θ j ) ostensibly has different coefficient functions which may vary with j , butobserve we may rewrite ˙ θ j = ω + (cid:16) N N P i =1 e iθ i (cid:17) cos θ j + (cid:16) iN N P i =1 e iθ i (cid:17) sin θ j . If the functions are invariant underpermutations of the θ j , we say the system is symmetric, like the aforementioned example.Lohe describes how to incorporate different amplitudes, frequencies, and phase shifts into the Kuramotoequations which ultimately amount to affine changes of variables. For simplicity, we try to leave out theseparameters.Strogatz and Watanabe proved the existence of N − S ) N , which is difficultto study directly. But by restricting to the submanifold where the N − λ ijkl = | e iθi − e iθk || e iθi − e iθl | | e iθj − e iθk || e iθj − e iθl | . There are (cid:0) N (cid:1) cross ratios, but it turnsout only N − t = 0 remain incident for all time (this is trivial) but bodies whichare not incident t = 0 will never collide because a nonzero cross ratio will always stay nonzero. A concise yet thorough textbook of Riemannian Geometry accessible to pure and applied mathematicians isJohn Lee’s
Riemannian Manifolds: An Introduction to Curvature [4].A
Riemannian metric on a smooth manifold M is a section of the tensored cotangent bundle ( T ∗ ) ⊗ M that is nondegenerate at every fiber. Informally, a Riemannian metric smoothly varies the inner productbetween tangent spaces. Two Riemannian manifolds are conformal if there is a diffeomorphism which scaleseach inner product by a positive factor. Conformal maps preserve angles and local orientation, but couldpossibly alter global geometric properties such as lengths of curves.The most common nontrivial conformal map is stereographic projection of a sphere S d to R d +1 . Embed S d in R d +1 in the standard way, which is via the subset { x ∈ R d +1 | | x | = 1 } . Let N = (0 , , . . . , ∈ S d be the north pole. For each x ∈ S d − N , draw a line from N to x and map x to the intersection of this2ine with the x d +1 = 0 plane, which we identify with R d . If we want to rigorously define the conformalmap, we compactify R d with one point and map N to it. A critical observation is that the map acts asthe identity on the equator, which is diffeomorphic to S d − . The stereographic projection also maps thesouthern hemisphere to the interior of the unit disc in R d and the northern hemisphere to the exterior of theunit disc. Showing sterographic projection is conformal is an elementary exercise in differential geometry, aproof of which is in Lee’s book.Since stereographic projection acts as the identity on the equator, the image of a Kuramoto model which takesplace on the equator is invariant under this transformation, even with respect to distances. We will see thata generalized Kuramoto model on S d is described by a path of conformal maps on the ambient space R d +1 ,which we can then apply the inverse stereographic projection to get a path of conformal transformations onthe sphere S d +1 .In this paper, we concern ourselves with conformal maps of spheres, with the Riemannian metric inherited bythe standard embedding. Rigid rotations of the sphere are conformal, but there are many others. Fractionallinear transformations of the form F ( z ) = az + bcz + d , with ac − bd > C ∼ = S . The formula is applied on the extended complex plane, but can be visualized as diffeomorphisms onthe sphere via the inverse stereographic projection. The group of fractional linear transformations is called the Mobius Group . We saw that the cross ratios between bodies on the Kuramoto model is preserved by its flow. A standardresult in complex analysis states that the maps which preserve cross ratios are the Mobius transformations.The space of the Kuramoto model is S , which we can embed in the standard way into C , and its imageunder the inverse stereographic projection is the equator of S .Mirollo et. al. saw that the flow of a Kuramoto model can be described by a dynamical system on the Liegroup of Mobius transformations. However, this dynamical system did not lie in the entire Lie group, butrather the open subgroup of Mobius transformations which preserve the unit disc and its boundary. If weapply the inverse stereographic projection, these are the conformal transformations of S which preservethe southern hemisphere. These transformations take the form M ζ,w ( z ) = ζ ( w − z − wz ), where | ζ | = 1 and | w | <
1. Hence, this group is diffeomorphic to the filled in torus S × D . This means if θ i is a body on theKuramoto model, its trajectory is modeled by θ i = M ζ,w ( θ i (0)), where ζ and w are parameters which aretime dependent, but apply for all bodies.Consider the one dimensional Kuramoto Model defined by A, B and C as mentioned in the beginning. Let a = − C + iB . Then Mirollo et. al. wrote down the explicit equations on S × D in the ( ζ, w ) coordinatesin terms of our given data by differentiating the trajectory equation θ i = M ζ,w ( θ i (0)) with respect to time.Specifically, ˙ w = −
12 (1 − | w | ) ζa ( p )˙ ζ = iA ( p ) ζ −
12 ( wa ( p ) − wa ( p ) ζ ) . N dynamical system on ( S ) N to just a three dimensional system, independent of N . Thisis a much more tractable system of equations! The generalized Kuramoto model with N bodies x , x , . . . , x N ∈ S d takes the form ˙ x i = Ω x i + X −h X, x i i x i ,where Ω is a predetermined antisymmetric matrix independent of time and i , and X ∈ R d is also independentof i , but possibly time dependent. Lohe saw the cross ratios λ ijkl = | x i − x k || x i − x l | x j − x k || x j − x l | are preserved. Thesequantities are preserved by Mobius transformations on ˆ R d +1 ∼ = S d +1 which map S d to itself. By derivinga formula for the Mobius transformations of this form and then differentiating with respect to time, Lohederived the reduced dynamical system. For completeness, we provide a derivation of the formula for thesegeneral Mobius transformations, which is conspicuously absent in many texts on conformal geometry.Liouville’s Theorem from conformal geometry (which is unrelated to his theorem on bounded entire functions)states that for d ≥
3, conformal mappings on ˆ R d +1 take the form f ( x ) = b + αA ( x − a ) | x − a | ε , where a, b ∈ R d +1 , α ∈ R , A ∈ SO + ( d + 1), and ε = 0 or ε = 2. As mentioned before, the mappings which describe theKuramoto model do not range over all possible conformal mappings, but rather are restricted to those whichmap S d to itself.Rather than directly applying Liouville’s Theorem, we can take the parametrization of the desired conformalmaps in the d = 2 case and generalize. Recall that Mobius transformation preserving the unit disc in R hasthe form M ζ,w ( z ) = ζ (cid:0) w − z − wz (cid:1) , | w | < z − w in the numerator instead, which means the derivations here can have asimilar, but different form as those which appear in other works. Navigating these conventions ultimatelyboils down to a transition map on coordinate charts describing the same manifold. Observe we have aLorentz boost determined by the w parameter, followed by a rotation of the ζ parameter. All conformalmappings can be decomposed as a product of this form. In higher dimensions, we replace multiplicationby ζ with the rotation determined by a parameter in SO ( d ), which we still call ζ , which we know how toparametrize in theory (take the matrix exponentials of the vector space of antisymmetric d × d matrices).So we will focus on the Lorentz boost.As with most complex fractions, we multiply the numerator and denominator by the denominator’s complexconjugate. w − z − wz · − wz − wz = w − w z + w | z | | − wz | With the denominator, we can rewrite | − wz | = h − wz, − wz i = h , i − h , wz i + h wz, wz i R , so 1 actuallydenotes the vector (1 ,
0) and the inner product is the standard one on R , not the Hermitian product becausethe components are real, not complex. Writing w = w + iw , z = z + iz , we can see h (1 , , wz i = Re( wz )= Re(( w − iw )( z + iz ))= w z + w z = h w, z i . Thus, h (1 , , (1 , i − h , wz i + h wz, wz i = 1 − h w, z i + | wz | = 1 − h w, z i + | w | | z | = | w − z | + (1 − | w | )(1 − | z | ) . For the denominator to be zero, we must have w = z and | w | = | z | = 1, which cannot happen because | w | < w − w z − z + w | z | = w − w z − z + w | z | − | w | ( − z + w | z | ) + | w | ( − z + w | z | )= ( − z + w | z | )(1 − | w | ) + w − w z − z | w | + w | w | | z | . The ultimate goal is to factor out a w in the latter four terms, and the remainder just so happens to equalthe denominator, leading to a nice cancellation. Notice that w − w z − z | w | + w | w | | z | = w − w z − zww + w | w | | z | = w (1 − wz − zw + | w | | z | ) . Further observe wz + zw = wz + wz = 2Re( wz )= 2 h w, z i . So cancellation yields a single w summand, leading us to conclude the vector form of Mobius transformationsappearing in Lohe’s paper: M ζ,w ( z ) = ζ (cid:16) ( − z + w | z | )(1 − | w | )1 − h w, z i + | w | | z | + w (cid:17) .
5n the following, we ignore ζ and focus on the Lorentz boost. If we substitute z ′ = − z | z | , we get M w ( z ) = ( w + z ′ )(1 −| w | ) | w + z ′ | + w . This substitution simplifies the formula whilst allowing M w = I for w = 0. If z = 0,there is a removable discontinuity and we can see M w (0) = w . We now show that M ζ,w maps the unit ballto itself. Computational assistance provided by Blue [1].Observe | M w ( z ) | = h M w ( z ) , M w ( z ) i = | w | + 2 (1 − | w | ) | w + z ′ | h w, w + z ′ i + (cid:16) − | w | | w + z ′ | (cid:17) | w + z ′ | . Clearing out denominators, we get | w − z ′ | | M w ( z ) | = | w | | w − z ′ | + 2(1 − | w | ) h w, w + z ′ i + (1 − | w | ) = | w | ( | w | + 2 h w, z ′ i + | z ′ | ) + 2(1 − | w | )( | w | + h w, z ′ i ) + (1 − | w | + | w | )= 2 h w, z ′ i + 1 + | w | | z | (a ton of things cancel)= (2 h w, z ′ i + | w | + | z ′ | ) + (1 + | w | | z ′ | − | w | − | z | )= | w + z ′ | + (1 − | w | )(1 − | z ′ | ) . Therefore, we have that | M w ( z ) | − (1 −| w | )(1 −| z ′ | ) | w + z ′ | . From the assumptions that | z | ≤ | w | <
1, wehave that the numerator of the righthand side is nonpositive. Therefore, | M w ( z ) | ≤
1, with equality when | z | = 1.This generalization is well-studied in conformal geometry, and it satisfies many of the same properties of theclassical Mobius transformations in two dimensions. The proofs that these maps respect properties enjoyedby the Mobius transformations ultimately comes down to more high school algebra. Proofs can be found inmany sources, such as Stoll’s text. From now on, we will use the variable R instead of ζ to emphasize thefact that it is a rotation matrix.The properties we use are: • The maps M R,w ( z ) are diffeomorphisms of the open unit ball and its boundary sphere in R d . • These maps preserve the sphere metric on the unit disc induced by its inclusion in to the southernhemisphere of the sphere. • These maps also preserve the cross ratios λ ijkl = | z i − z k || z i − z l | | z j − z k || z j − z l | . (Here z i refers to a body on the sphere S d , not a vector component). In the two dimensional case, division by vectors makes sense because ofthe complex structure, but in general we have to take ratios of lengths. • The inverse of M R,w is ( M − w ◦ R T ) = M R T ,Rw . Recall that R is orthogonal, so R − = R T . • All maps with the above properties can be realized in the above form, so SO ( d ) × D d is indeed aparametrization of the group of conformal maps on S d +1 which preserve S d . • SO ( d ) × D d is isomorphic to the Lorentz group SO + ( d, Lohe’s Reduced Equations
The generalization of the Kuramoto Model of a collection of coupled bodies { x i } on S d is ˙ x i = Ω x i + X −h x i , X i x i , where Ω is an antisymmetric frequency matrix and X determines how the bodies are coupled.Lohe has shown this system has conserved quantities λ ijkl = | x i − x k || x i − x l | | x j − x k || x j − x l | , which know are preserved bygeneralized Mobius transformations.Hence, the trajectory of a body x i is x i ( t ) = M ζ ( t ) ,w ( t ) ( x i (0)), and we can apply the same Mobius transforma-tion to every body to determine the state of the system at a given time. This reduces our dynamical systemon ( S d ) N with dimension N d to a much simpler dynamical system on a Lie group of dimension ( d +1)+ ( d − d ,which is independent of N . We are free to reparametrize SO ( d ) and D d +1 to time-dependent coordinateswhich happen to induce even nicer looking conserved quantities.Let u i ( t ) = M w ( t ) ( x i ( t )). If we assume ˙ w = Ω w + (1+ | w | ) X −h w, X i w , we have h u i ( t ) , u j ( t ) i as a conservedquantity for all i and j . This is the so-called “Watanabe-Strogatz” transform which changes coordinates suchthat the bodies appear to be moving under standard rigid motions of the sphere. This allows us to considerthe rotation parameter separately. Such a w can be derived by assuming the inner product is preserved andworking backwards to find the constraint on w . This equation is independent of N , so taking N → ∞ givesEd Ott’s differential equation for probability distribution on spheres. Thus, we have that u i is modeled by atrajectory of the form u i ( t ) = R ( t ) u i (0), where R ( t ) is a path in SO ( d ). Without loss of generality, we mayassume R (0) = I via a Lie group translation.Observe that ˙ u i = ˙ Ru i (0) = ˙ RR T u i ( t ). Let A = ˙ RR T − Ω. The tangent space of SO ( d ) is the space ofantisymmetric matrices, so ˙ R is antisymmetric. Differentiating the identity RR T = I , we can see ˙ RR T = − R T ˙ R = ( ˙ RR T ) T , so we can conclude A is also antisymmetric. Now we have the equation differential˙ R = (Ω + A ) R , and Lohe shows that choosing A ij = X i w j − X j w i satisfies ˙ RR T u i = ( A + Ω) Ru i , which canbe verified via a computer algebra system. If we can solve for R and w , we can then write the Mobius transformations in the following time dependentmatrix form: M R,w = (cid:20) R f ( ~w ) − f ( ~w ) T (cid:21) where f is a canonical transformation of w to a vector in R d with a well-known formula. Then if we want totrack the movement of a body over time, we apply this time dependent matrix to its initial position.Max Lohe did the labor of actually deriving the reduced equations, but what we have done is reinterpret thisproblem as a question about dynamical systems on conformal mapping groups. This is a mathematically nicereorganization, but in order to prove its worth, someone needs to use insights from dynamics on SO ( d,
1) andconformal geometry to deduce some qualitative behavior on a specific instance of the generalized Kuramotosystem. Since the stereographic projection acts as the identity on the equatorial sphere, we simultaneouslyhave two topologically equivalent vector fields which we can visualize on R d +1 and S d +1 . The theory of7ector fields on spheres comes with many topological obstructions, and it is my hope that somehow theseobstructions can be used to deduce qualitative behavior of the Kuramoto model. References [1] Blue,
Finding the norm of w + −| w | | w − z | ( w − z ) , where w and z are in R n . Mathematics Stack Exchange,2018. URL: http://math.stackexchange.com/questions/3023257 [2] S. Chandra, M. Girvan, and E. Ott
Complexity Reduction for Systems of Interacting Orientable Agents:Beyond The Kuramoto Model . ArXiv preprint, 2018.[3] B. Chen, J. Engelbrecht, R. Mirollo
Hyperbolic geometry of Kuramoto oscillator networks . J. Phys. A:Math. Theor., 2017.[4] J. Lee
Riemannian Manifolds: An Introduction to Curvature . Springer, 1997.[5] M. Lohe
Higher-dimensional generalizations of the Watanabe Strogatz transform for vector models ofsynchronization . J. Phys. A: Math. Theor., 2018.[6] M. Stoll
Harmonic and Subharmonic Function Theory on the Harmonic Ball . Cambridge UniversityPress, 2016.[7] S. Watanabe, S. Strogatz